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 Question from Brad: I found this on your site. Question: what is the sum of the first 100 whole numbers?? Is there a different formula if the numbers begin at a number other than one? For example What is the series I want to add is goes from 7 - 53?

You can use the same technique that is attributed to Gauss and used in our previous response you mentioned. That is call the sum $S$ and write twice the sum as

2S =   7 +  8 +   9 + ... + 53 +
53 + 52 + 51 + ... +  7

2S = 60 + 60 + 60 + ... + 60

which is 60 times the number of terms. How many terms are there? 53 - 7 = 46 hence you start with 7 and add 1, 46 times to get to 53 and hence you have 7 and then 46 more terms so there are 47 terms. Thus

2S = 47 times 60

or $S = \large \frac12 \normalsize 60 \times 47.$

You could also use the formula S = n[2a+(n-1)d]/2 which is given in the resource you quote but I prefer to remember

S = (the first term + the last term) times (the number of terms) divided by 2.

All three of these techniques apply to a more general situation also. Both the sum of the first 100 whole numbers and your sequence start with a number and then add 1 to ge the second term, add 1 again to get the third term and so on. What is the number you add each time is not 1 but something else? For example start with 5 and add 3 to get the second term. Then add 3 again to ge the third term and so on.

5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29

What is this sum?

Harley